IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v23y2021i1d10.1007_s11009-020-09830-w.html
   My bibliography  Save this article

On Rereading Stein’s Lemma: Its Intrinsic Connection with Cramér-Rao Identity and Some New Identities

Author

Listed:
  • Nitis Mukhopadhyay

    (University of Connecticut)

Abstract

Now is an opportune time to revisit Stein’s (1973) beautiful lemma all over again. It is especially so since researchers have recently begun discovering a great deal of potential of closely related Stein’s unbiased risk estimate (SURE) in a number of directions involving novel applications. In recognition of the importance of the topic of Stein’s (1973; Ann Statist 9:1135–1151, 1981) research and its elegance, we include a selective review from the field. The process of rereading Stein’s lemma and reliving its awesome simplicity as well as versatility rekindled a number of personal thoughts, queries, and observations. A number of new and interesting insights are highlighted in the spirit of providing updated and futuristic versions of the celebrated lemma by largely focusing on univariate continuous distributions not belonging to an exponential family. In doing so, a number of new identities have emerged when the parent population is continuous, but they are highly non-normal. Last, but not the least, we have argued that there is no big foundational difference between the basic messages obtained via Stein’s identity and Cramér-Rao identity.

Suggested Citation

  • Nitis Mukhopadhyay, 2021. "On Rereading Stein’s Lemma: Its Intrinsic Connection with Cramér-Rao Identity and Some New Identities," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 355-367, March.
  • Handle: RePEc:spr:metcap:v:23:y:2021:i:1:d:10.1007_s11009-020-09830-w
    DOI: 10.1007/s11009-020-09830-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-020-09830-w
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s11009-020-09830-w?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. V. Papathanasiou, 1995. "A characterization of the Pearson system of distributions and the associated orthogonal polynomials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(1), pages 171-176, January.
    2. Landsman, Zinoviy & Vanduffel, Steven & Yao, Jing, 2015. "Some Stein-type inequalities for multivariate elliptical distributions and applications," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 54-62.
    3. A. Mukherjee & K. Chen & N. Wang & J. Zhu, 2015. "On the degrees of freedom of reduced-rank estimators in multivariate regression," Biometrika, Biometrika Trust, vol. 102(2), pages 457-477.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Luo, Chongliang & Liang, Jian & Li, Gen & Wang, Fei & Zhang, Changshui & Dey, Dipak K. & Chen, Kun, 2018. "Leveraging mixed and incomplete outcomes via reduced-rank modeling," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 378-394.
    2. Matsuda, Takeru & Strawderman, William E., 2019. "Improved loss estimation for a normal mean matrix," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 300-311.
    3. Moawia Alghalith & Wing-Keung Wong, 2020. "Extension of Stein's Lemmas to General Functions and Distributions," Advances in Decision Sciences, Asia University, Taiwan, vol. 24(4), pages 77-88, December.
    4. James Younker, 2022. "Calculating Effective Degrees of Freedom for Forecast Combinations and Ensemble Models," Discussion Papers 2022-19, Bank of Canada.
    5. Yang, Yuehan & Xia, Siwei & Yang, Hu, 2023. "Multivariate sparse Laplacian shrinkage for joint estimation of two graphical structures," Computational Statistics & Data Analysis, Elsevier, vol. 178(C).
    6. Guillaume Allaire Pouliot & Zhen Xie, 2022. "Degrees of Freedom and Information Criteria for the Synthetic Control Method," Papers 2207.02943, arXiv.org.
    7. Hansen, Niels Richard, 2018. "On Stein’s unbiased risk estimate for reduced rank estimators," Statistics & Probability Letters, Elsevier, vol. 135(C), pages 76-82.
    8. Shushi, Tomer, 2018. "Stein’s lemma for truncated elliptical random vectors," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 297-303.
    9. Giorgos Afendras, 2013. "Unified extension of variance bounds for integrated Pearson family," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(4), pages 687-702, August.
    10. Goh, Gyuhyeong & Dey, Dipak K. & Chen, Kun, 2017. "Bayesian sparse reduced rank multivariate regression," Journal of Multivariate Analysis, Elsevier, vol. 157(C), pages 14-28.
    11. Christophe Ley & Gesine Reinert & Yvik Swan, 2014. "Approximate Computation of Expectations: the Canonical Stein Operator," Working Papers ECARES ECARES 2014-36, ULB -- Universite Libre de Bruxelles.
    12. Kohei Yoshikawa & Shuichi Kawano, 2023. "Sparse reduced-rank regression for simultaneous rank and variable selection via manifold optimization," Computational Statistics, Springer, vol. 38(1), pages 53-75, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:metcap:v:23:y:2021:i:1:d:10.1007_s11009-020-09830-w. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.